Clumath343s2ch5s1examples

Question

Explain why a 2x2 matrix can have at most two distinct eigenvalues?

Asnwer

If a 2x2 matrix A were to have three distinct eigenvalues, then by Theorem 2 (If $v_1, ..., v_r$ are eigenvectors that correspond to distinct eigenvalues $\lambda_1,...,\lambda_2$ of an $n \times n$ matrix $A$, then the set ${v_1, ..., v_r}$ is linearly independent) there would correspond 3 linearly independent eigenvectors (one for each eigenvalue). This is impossible because the vectors all belong to a 2-dimensional vector space, in which any set of 3 vectors is linearly dependent.

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